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Top1. Introduction
We understand Territorial Design (TD) or zone design as the grouping of small geographical areas or units into zones that are acceptable according to the requirements imposed by the problem under study. Depending on the context, these requirements can include the generation of connected zones that contain about the same amount of habitants, clients, communication means, public services, etc. The zone design has several applications such as the creation of school zones, zones with the appropriate characteristics for socioeconomic analysis, sales, maintenance or services territories design, geographical design for census and logistic applications. Diverse interesting TD applications can be seen in Ricca et al. (2008), Rios-Mercado et al. (2007), Salazar et al. (2011) and Salazar et al. (2011a). In the geographical aggregation process in TD, hierarchical or partitioning clustering methods adequate to the problem are often employed.
For the aggregation done in this work, the elements grouped belong to the geographical territorial unit known as Basic Geo-statistical Area (AGEB for its initials in Spanish), which is defined as the minimum geographical division used for census and statistical purposes in the National Institute of Geography and Informatics INEGI for its initials in Spanish (INEGI, 2011). INEGI is an autonomous entity of the Mexican government, dedicated to the coordination of the National Informatics Statistics and Geography System and is the institution in charge to do the population census every ten years. Its main goal is to achieve the provision of opportune, verified, relevant and quality information to the public and the State, with the aim of contributing to the national development, under the accessibility, transparency, objectivity and independency principles (INEGI, 2011).
The groups generated by the partitioning we have developed are formed by a set of AGEBs and due to the partitioning properties; the compactness characteristic of TD is implicitly satisfied. This territorial partitioning is previously necessary to solve the location allocation problem (LAP) for a TDP with dense demand, which is the main purpose of this work: finding the geographical coordinates (longitude, latitude) of the location of a distribution center (DC) that provides services to a group of communities that are found in every AGEB, where each AGEB is represented by its centroid. The location should be the one that minimizes the travelling expenses by finding the geographical coordinates of the center of the centroids. The populations from these communities represent the potential clients of the DC, and their demand is modeled continuous demand functions with two variables based on population density of every group (Newling, 1969).
Due to the numeric nature of the solutions obtained, this problem addresses a continuous case of LAP. Additionally, with the mathematical approach associated, we make use of a geographical information system (GIS) with the goal of creating maps (Zamora, 2006).